A numerical semigroup $S$ is a cofinite, additively closed subset of the nonnegative integers that contains $0$. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring ${\mathbb{𝔽}}_q[x]$ is zero for any finite field ${\mathbb{𝔽}}_q$; we prove that the numerical semigroup algebra ${\mathbb{𝔽}}_q[S]$ also has atomic density zero for any numerical semigroup $S$. We also examine the particular algebra ${\mathbb{𝔽}}_2[x^2,x^3]$ in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field ${\mathbb{𝔽}}_q$.

Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen–Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.

The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski-type inequality given in their paper. We also recover a result of Cutkosky, Srinivasan and Verma as a simple consequence of our inequality.

This article is a natural continuation of our previous works [Dutta 2021] and [Dutta 2022]. In this article, we employ similar ideas as in [Alexandru, Popescu and Zaharescu 1990] to provide an estimate of $IC(K(X)\mid K,v)$ when $(K(X)\mid K,v)$ is a valuation algebraic extension. Our central result is an analogue of [Dutta 2022, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for $v$ over $K$ in the setting of valuation algebraic extensions.

In 2011, Khurana, Lam and Wang defined the following property:

(*) A commutative unital ring $A$ satisfies the property “power stable range one” if for all $a,b\in A$ with $aA+bA=A$ there is an integer $N=N(a,b)\ge 1$ and $\lambda =\lambda (a,b)\in A$ such that $b^N+\lambda a\in A^{\times }$, the unit group of $A$.

In 2019, Berman and Erman considered rings with the following property:

(**) A commutative unital ring $A$ has enough homogeneous polynomials if for any $k\ge 1$ and set $S:=\{p_1,p_2,\dots ,p_k\}$, of primitive points in $A^n$ and any $n\ge 2$, there exists a homogeneous polynomial $P(X_1,X_2,\dots ,X_n)\in A[X_1,X_2,\dots ,X_n]$ with $\text{deg }P\ge 1$ and $P(p_i)\in A^{\times }$ for $1\le i\le k$.

We show that the two properties (*) and (**) are equivalent and we shall call a commutative unital ring with these properties a good ring.

When $A$ is a commutative unital ring of pictorsion as defined by Gabber, Lorenzini and Liu in 2015, we show that $A$ is a good ring. Using a Dedekind domain built by Goldman in 1963, we show that the converse is false.

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras $C$ over a base $A$ with fiber $B$. The free extensions are deformations of the usual tensor product; when $C$ is also Gorenstein, so are $A$ and $B$, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator $F$ for $C$ as a homogeneous “polynomial” in $T$ and the dual variables for $B$, and given the dual generator for $B$, we give sufficient conditions on $F$ that ensure that $C$ is a free extension of $A=\text{k}[t]∕(t^n)$ with fiber $B$. We give examples exploring the sharpness of the statements. We also consider a special set of coinvariant algebras $C$ which are free extensions of $A$, but which do not satisfy the sufficient conditions of our main result.

We consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines (points of multiplicity $i$) coincide. We show that in one of these examples the containment $I^{(3)}\subseteq I^2$ holds, whereas it fails in the other. We also show that the containment fails for a subarrangement of $21$ lines. The interest in the containment relation between $I^{(3)}$ and $I^2$ for ideals of points in ${\mathbb{P}}^2$ is motivated by a question posted by Hochster and Huneke in $2002$. Configurations of points with $I^{(3)}⊈I^2$ are quite rare. Our example reveals two particular features: All points are defined over $\mathbb{Q}$ and all intersection points of lines are involved. In examples studied by now only points with multiplicity $i\ge 3$ were considered. The novelty of our arrangements lies in the geometry of the element in $I^{(3)}$ which witness the noncontainment in $I^2$. In all previous examples such an element was a product of linear forms. Now, in both cases there is an irreducible curve of higher degree involved.

Let $R$ be a commutative Noetherian ring of dimension $d$ and $M$ a commutative cancellative torsion-free seminormal monoid. Then: (1) Let $A$ be a ring of type $R[d,m,n]$ and $P$ be a projective $A[M]$-module of rank $r\ge \text{max }\{2,d+1\}$. Then the action of $E(A[M]\oplus P)$ on $(A[M]\oplus P)$ is transitive and (2) Assume $(R,m,K)$ is a regular local ring containing a field $k$ such that either $\text{char }k=0$ or $\text{char }k=p$ and $\text{tr}$-$\text{deg }K∕{\mathbb{𝔽}}_p\ge 1$. Let $A$ be a ring of type $R{[d,m,n]}^{\ast }$ and $f\in R$ be a regular parameter. Then all finitely generated projective modules over $A[M]$, $A{[M]}_f$ and $A[M]{\otimes }_{R}R(T)$ are free. When $M$ is free both results are due to Keshari and Lokhande (2014).

We provide a formula to compute the big Cohen–Macaulay test ideal for triples $((R,\mathrm{\Delta }),{\mathfrak{𝔞}}^t)$ where $R$ is a mixed characteristic toric ring and $\mathfrak{𝔞}$ is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero.